It seems that Overlay does only work when tracking-script is /js/piwik.js.

After clicking the overlay button, client.js can not be served (404-error).

After reading /js/README I decided to embed the tracking script via /js/index.php for some reason.
Overlay isn't a feature I use very frequently - it seemed broken, I blamed Apache/nginx-rewriting for this and planned to investigate deeper as soon as i got some spare time.
When the error also appeared on a second installation with completely different environment, i finally wanted to fix the problem.

The first request to overlay mode seems to load a blank page with just minimized Javascript:/index.php?module=Overlay&action=notifyParentIframe

This script seems to build the framed page and tries to load client.js: